We consider random number conversion (RNC) through random number storage with restricted size. We clarify the relation between the performance of RNC and the size of storage in the framework of the first- and second-order asymptotics, and derive their rate regions. Then, we show that the results for RNC with restricted storage recover those for conventional RNC without storage in the limit of storage size. To treat RNC via restricted storage, we introduce a new kind of probability distributions named generalized Rayleigh-normal distributions. Using the generalized Rayleigh-normal distributions, we can describe the second-order asymptotic behavior of RNC via restricted storage in a unified manner. As an application to quantum information theory, we analyze LOCC conversion via entanglement storage with restricted size. Moreover, we derive the optimal LOCC compression rate under a constraint of conversion accuracy.
We consider random number conversion (RNC) through random number storage with restricted size. We clarify the relation between the performance of RNC and the size of storage in the framework of the first- and second-order asymptotics, and derive their rate regions. Then, we show that the results for RNC with restricted storage recover those for conventional RNC without storage in the limit of storage size. To treat RNC via restricted storage, we introduce a new kind of probability distributions named generalized Rayleigh-normal distributions. Using the generalized Rayleigh-normal distributions, we can describe the second-order asymptotic behavior of RNC via restricted storage in a unified manner. As an application to quantum information theory, we analyze LOCC conversion via entanglement storage with restricted size. Moreover, we derive the optimal LOCC compression rate under a constraint of conversion accuracy.
We consider the optimal approximate conversion between multiple copies of pure entangled states in quantum systems when only local operations and classical communications (LOCC) are allowed. This problem contains a kind of cloning problem with LOCC restriction as a special case. To derive the asymptotic LOCC conversion rate, we consider two kinds of approximate conversions, deterministic conversion and majorization conversion, for probability distributions, and solve their asymptotic conversion rates up to the second order. Then, the asymptotic LOCC conversion rate is obtained via the natural relation between the LOCC conversion and the majorization conversion. To derive these asymptotic rates, we introduce new probability distributions named Rayleigh-normal distributions. The family of Rayleigh-normal distributions includes a Rayleigh distribution and coincides with the standard normal distribution in the limit case, and the optimal conversion rate is represented by Rayleigh-normal distribution in a unified manner.
We discuss the asymptotic behavior of conversions between two independent and identical distributions up to the second-order conversion rate when the conversion is produced by a deterministic function from the input probability space to the output probability space. To derive the second-order conversion rate, we introduce new probability distributions named Rayleigh-normal distributions. The family of Rayleigh-normal distributions includes a Rayleigh distribution and coincides with the standard normal distribution in the limit case. Using this family of probability distributions, we represent the asymptotic second-order rates for the distribution conversion. As an application, we also consider the asymptotic behavior of conversions between the multiple copies of two pure entangled states in quantum systems when only local operations and classical communications (LOCC) are allowed. This problem contains entanglement concentration, entanglement dilution, and a kind of cloning problem with LOCC restriction as special cases.
We study the problem of efficient compression of a stochastic source of probability distributions. It can be viewed as a generalization of Shannon's source coding problem. It has relation to the theory of common randomness, as well as to channel coding and rate--distortion theory: in the first two subjects ``inverses'' to established coding theorems can be derived, yielding a new approach to proving converse theorems, in the third we find a new proof of Shannon's rate--distortion theorem. After reviewing the known lower bound for the optimal compression rate, we present a number of approaches to achieve it by code constructions. Our main results are: a better understanding of the known lower bounds on the compression rate by means of a strong version of this statement, a review of a construction achieving the lower bound by using common randomness which we complement by showing the optimal use of the latter within a class of protocols. Then we review another approach, not dependent on common randomness, to minimizing the compression rate, providing some insight into its combinatorial structure, and suggesting an algorithm to optimize it. The second part of the paper is concerned with the generalization of the problem to quantum information theory: the compression of mixed quantum states. Here, after reviewing the known lower bound we contribute a strong version of it, and discuss the relation of the problem to other issues in quantum information theory.
Resource theories provide a universal framework for quantifying properties of quantum states, such as entanglement, magic, athermality, and asymmetry, as resources under various constraints encountered in quantum science. Since realistic settings typically impose several constraints at once, it is important to study multi-resource theories that combine them. Among these, combinations with asymmetry are especially natural, as symmetry is fundamental and ubiquitous in physics. Here we develop a general theory that treats combinations of finite-group asymmetry with a broad range of resource theories in a unified manner. Our key observation is that the effect of combining the two constraints is governed by the accessibility of symmetry information under the underlying resource theory: whether the information needed for the conversion can be extracted from the input and used to control free operations. If this can be done with asymptotically vanishing error, we show that the conversion rate of the combined theory equals the smaller of the rates of the two constituent theories, so combining them incurs no additional cost. This condition holds for LOCC on arbitrary mixed multipartite states, magic-state conversion of multiqubit systems under Clifford symmetries, and Gibbs-preserving operations. Thermal operations, being time-translation covariant, can access only limited symmetry information, giving an additional compatibility condition between the two symmetries. We also analyze symmetry imposed on elementary operations rather than only on the resulting channel. We establish analogous results for LOCC and thermal operations under suitable assumptions, and for magic when the symmetry is represented by Pauli operators. Finally, we show that finite-group symmetry does not reduce the asymptotic ergotropy per copy and hence introduces no new completely passive states.
Recently, two of the authors showed that entanglement concentration is irreversible. However, it is still not clear what kind of LOCC conversion is reversible. We derive the necessary and sufficient condition for reversibility of LOCC conversion between two bipartite pure entangled states in an asymptotic setting. Simultaneously, we evaluate how many copies of the initial state is to be lost to overcome irreversibility of LOCC conversion. Our result is useful for designing how to store entangled states via LOCC operations without error.
This dissertation investigates relative entropies, also called generalized divergences, and how they can be used to characterize information-theoretic tasks in quantum information theory. The main goal is to further refine characterizations of the optimal rates for quantum source coding, state redistribution, and measurement compression with quantum side information via second order asymptotic expansions and strong converse theorems. The dissertation consists of a mathematical and an information-theoretic part. In the mathematical part, we focus on the α-sandwiched Rényi divergence (α-SRD). We first investigate the limit α→0 to determine whether this recovers the well-known 0-Rényi relative divergence. We then prove various new results for entropic quantities derived from the α-SRD, including dimension bounds and useful bounds in terms of the fidelity between two quantum states. Furthermore, we derive a necessary and sufficient algebraic condition for equality in the data processing inequality (viz. monotonicity under quantum operations) for the α-SRD, and give applications to entropic bounds. In the information-theoretic part, we first derive the second order asymptotics of visible quantum source coding using a mixed source. For the achievability part, we develop universal quantum source codes achieving a given second order rate for a memoryless source. As a corollary of the main result, we obtain the second order asymptotics of quantum source coding using a single memoryless source. We then prove strong converse theorems for state redistribution (with or without feedback) and measurement compression with quantum side information. The key ingredients in proving these theorems are the aforementioned fidelity bounds on Rényi entropic quantities derived from the α-SRD.